Neutral atoms are entangled in hyperfine states via Rydberg blockade

Ions and neutral atoms held in electromagnetic traps are two of many candidates that may one day become the qubits in a quantum computer: Their hyperfine states could serve as the computer’s ones and zeroes. Ions interact via long-range Coulomb forces, which can facilitate creation of the entangled states that are the prerequisite for quantum computation. But that same Coulomb interaction gives rise to collective motions that can disrupt a qubit array. Atoms aren’t susceptible to such disruptions. But they’re also more difficult to entangle.

Last year two research groups independently demonstrated a long-range interaction, called Rydberg blockade, between trapped neutral atoms; they published their results in a pair of back-to-back papers. (See Physics Today, February 2009, page 15 .)

One group, led by Mark Saffman and Thad Walker at the University of Wisconsin-Madison, showed the blockade in its simplest form: When two atoms were separated by several microns, exciting one into a Rydberg state—an energetic state with a large, delocalized electronic wave-function—prevented the other from being similarly excited. ¹ The blockade works because the energy of two Rydberg atoms with respect to the ground state is less than twice the energy of one Rydberg atom, so the second Rydberg excitation is shifted out of resonance with the excitation laser. The other group, led by Philippe Grangier and Antoine Browaeys of the Université Paris-Sud, the Institute d’Optique, and CNRS, used Rydberg blockade to entangle two atoms, with one in the Rydberg state and the other in the ground state. ²

Again publishing their results back to back, both groups have now used Rydberg blockade to entangle pairs of atoms in two hyperfine levels of the atomic ground state. The Paris researchers did it by transforming their ground-Rydberg entangled state into a hyperfine-hyperfine entangled state. ³ The Wisconsin researchers constructed a quantum logic gate called a controlled-NOT, or CNOT, gate: a sequence of laser pulses, involving excitations to the Rydberg state, that changes the state of a target atom if and only if a control atom is in a particular hyperfine state. ⁴ Applying the CNOT gate when the control atom is in a superposition of states entangles the two atoms. A perfectly working CNOT gate, plus the ability to manipulate single qubits, can be the basis for all the qubit interactions that are needed in a quantum computer.

The Paris protocol

In their work last year, the Paris researchers blasted a pair of rubidium-87 atoms with a Rydberg-exciting laser pulse. Only one atom was excited, but the excitation was delocalized over the pair—that is, the pulse created a superposition of the two-atom states |0r〉 and |r0〉, where 0 is the ground state and r is the Rydberg state. Since such a superposition can’t be represented as a product of two wavefunctions, one localized on each atom, it is an entangled state. But the Rydberg states themselves aren’t suitable for use as qubits. Rydberg atoms aren’t confined by the optical traps, and they readily undergo spontaneous emission. More troubling, the entangled state was actually $\left(|\text{r}0〉+e^{i\varphi }|\text{0r}〉\right)/\sqrt{2}$ , where the phase ϕ depended on the positions of the atoms in their traps, which varied randomly from one repetition of the experiment to another. That uncontrollable variation hampered the researchers’ efforts to verify that the atoms were really entangled and made it impossible to exploit the entanglement.

Now they’ve added a second laser pulse that moves the Rydberg atom into a different ground-state hyperfine level—call it |1〉. That pulse also imparts a phase to the system, but since it stimulates an emission rather than an absorption, and since the atoms don’t move much in the 200 ns between the start of the first pulse and the end of the second, the two phases are nearly equal in magnitude and opposite in sign. To a good approximation, they create the symmetric superposition of |01 and |10〉.

The Madison method

The Wisconsin researchers considered two different pulse sequences for their CNOT gate. Figure 1 shows both, with the numbers indicating the order in which the pulses are applied. Most of the pulses are π pulses, which have exactly the right duration to move an atom from one state to the other. There is also a 2π pulse (which returns the atom to its initial state and imparts a π phase shift to it) and π/2 pulses (which can leave the atom in a superposition of states).

Figure 1a shows a CNOT sequence based on Hadamard (essentially π/2) pulses plus a controlled π rotation about the z axis, or H-C_z CNOT. One can imagine it applied to control and target atoms that are both initially in state |1〉. The first pulse moves the target atom into a superposition of states |1〉 and |0〉. The second excites the control atom to a Rydberg state |r〉. The third, applied to the target atom, has no effect because of Rydberg blockade. The fourth returns the control atom to |1〉, and the fifth, acting on a superposition of states, completes the target atom’s journey to |0〉.

However, if the control atom starts in |0〉, pulse 2 is of the wrong energy to raise it to |r〉. So pulse 3 takes the |1〉 component of the target atom on a round trip to |r〉 and back and changes its phase, which means that pulse 5 returns the atom to |1〉 rather than lowering it to |0〉. Similarly, if the target atom starts in |0〉, its state is flipped if and only if the control atom starts in |1〉. The other CNOT gate, shown in figure 1b, is the controlled amplitude swap, or A-S. It accomplishes the same thing using only π pulses.

Applying a CNOT gate when the control atom is in a superposition of states produces an entangled state. For example, when the control atom is in $\left(|0〉+i|\text{1}〉\right)/\sqrt{2}$ and the target is in |0〉, the CNOT gate yields $\left(|01〉+|\text{10}〉\right)/\sqrt{2}$ , the same as the Paris researchers produced. Different initial conditions yield other entangled states.

Entangled webs we weave

“The most challenging part,” says the Paris group’s Browaeys, “was to analyze the amount of entanglement we produced in the experiment.” To determine how faithfully their schemes produced the desired entangled state $\left(|01〉+|\text{10}〉\right)/\sqrt{2}$ , both groups started by repeatedly preparing the state and measuring the states of both atoms. Usually (but not always, because the experiments were imperfect), when the first atom was in |0〉, the second was in |1〉, and vice versa. But that observation is not enough to prove entanglement. The atoms could have been produced half the time in the unentangled state |01〉 and half the time in |10〉.

A quantum statistical ensemble is characterized by the density matrix ρ. The matrix’s diagonal elements ρ _k,k are the probabilities P_k of finding the system in state k, where in this case k is one of the four pure states |11〉, |10〉, |01〉, and |00〉. The off-diagonal element relevant to verifying the entangled state $\left(|01〉+|\text{10}〉\right)/\sqrt{2}$ is ρ_10,01. Called the coherence between |10〉 and |01〉, it’s a measure of how tightly the phase is constrained between those two components.

The fidelity, F = (P ₀₁ + P₁₀)/2 + ρ_10,01, quantifies both how close a process comes to reliably producing the state $\left(|01〉+|\text{10}〉\right)/\sqrt{2}$ and how much entanglement it produces. When F 〉 $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ the atoms are entangled in the sense that their state can’t be represented as a statistical ensemble of products of local wavefunctions. (An accessible proof is given in reference .) There’s a second threshold at $F>\sqrt{2}/2$ , corresponding to violation of Bell’s inequalities.

In an experiment that entangles the spins of two particles, off-diagonal elements can be measured, and entanglement verified, by measuring the spins in multiple directions. A process that prepares the state $\left(|\uparrow \downarrow 〉+|\downarrow \uparrow 〉\right)/\sqrt{2}$ , where ↑ and ↓ are the spins in the z direction, produces particles that are opposite in spin in every direction. But if the process randomly produces either |↑↓〉 or |↓↑〉, the spins measured in the x direction would appear uncorrelated.

The two-state system that comprises the relevant hyperfine states is mathematically equivalent to a spin- $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ particle. Coherent pulses of light do the equivalent of rotating the spins to allow them to be probed in different directions. The Wisconsin researchers applied π/2 pulses to both atoms with a variable delay between them; the Paris group irradiated both atoms with a single pulse of variable duration. In both cases, ρ_10,01 was revealed as the amplitude of oscillation of the parity signal π = P ₀₀ + P ₁₁ − P ₁₀ − P ₀₁ as a function of that delay or duration. Figure 2 shows the oscillation for the Paris group’s experiment.

The Wisconsin group found that their best results came from the H-C_z CNOT gate, which prepared states with a fidelity of F = 0.48 ± 0.06, just below the threshold for entanglement. The Paris group measured a fidelity of F = 0.46 ± 0.06. But both groups’ atoms escaped their traps a significant fraction of the time—17% for the Wisconsin group and 39% for the Paris group—so the measured probability for the system to be in any state was less than one. (That’s a problem that experimenters who work with ions just don’t have to worry about, since loss from ion traps is negligible.) Both groups therefore normalized their results to give the fidelity for only those repetitions of the experiment in which no atoms were lost. For that a posteriori entanglement fidelity, the Wisconsin researchers obtained 0.58, the Paris researchers 0.75.

Both groups are working on optimizing their experiments—stabilizing their lasers, further cooling the atoms within their traps, and improving their vacuum systems—in order to suppress atom loss and increase fidelity. In addition, the Wisconsin researchers have their sights on the multiqubit entanglement necessary for basic quantum computing. Says Saffman, “A primary goal for the next five years or so is running quantum programs on 10 to 20 qubits and studying error correction.”